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Properties

Label 2-864-72.13-c1-0-9
Degree $2$
Conductor $864$
Sign $0.218 + 0.975i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 + 0.687i)5-s + (−1.80 − 3.12i)7-s + (1.83 − 1.05i)11-s + (−0.887 − 0.512i)13-s − 0.808·17-s − 7.43i·19-s + (−1.65 + 2.86i)23-s + (−1.55 − 2.69i)25-s + (7.71 − 4.45i)29-s + (−3.26 + 5.65i)31-s − 4.96i·35-s − 4.01i·37-s + (3.45 − 5.99i)41-s + (0.245 − 0.142i)43-s + (−3.61 − 6.25i)47-s + ⋯
L(s)  = 1  + (0.532 + 0.307i)5-s + (−0.682 − 1.18i)7-s + (0.552 − 0.319i)11-s + (−0.246 − 0.142i)13-s − 0.196·17-s − 1.70i·19-s + (−0.345 + 0.597i)23-s + (−0.310 − 0.538i)25-s + (1.43 − 0.826i)29-s + (−0.586 + 1.01i)31-s − 0.839i·35-s − 0.660i·37-s + (0.540 − 0.935i)41-s + (0.0375 − 0.0216i)43-s + (−0.527 − 0.912i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09438 - 0.876773i\)
\(L(\frac12)\) \(\approx\) \(1.09438 - 0.876773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-1.19 - 0.687i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.80 + 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.83 + 1.05i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.887 + 0.512i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.808T + 17T^{2} \)
19 \( 1 + 7.43iT - 19T^{2} \)
23 \( 1 + (1.65 - 2.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.71 + 4.45i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.26 - 5.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.01iT - 37T^{2} \)
41 \( 1 + (-3.45 + 5.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.245 + 0.142i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.61 + 6.25i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.86iT - 53T^{2} \)
59 \( 1 + (-7.06 - 4.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.31 + 3.64i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.43 - 1.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.69T + 71T^{2} \)
73 \( 1 - 0.409T + 73T^{2} \)
79 \( 1 + (-0.0456 - 0.0790i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.40 - 1.39i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 + (2.76 + 4.78i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01628162013063652885989006246, −9.308858708203183319138722843159, −8.349692529956442035977686593974, −7.08235854704900901545459062943, −6.76529168807736006894393757974, −5.70610644183958677200011713211, −4.49462582769803669601105600711, −3.55193363643982258665601324663, −2.40603385028773892010241186636, −0.68743857956608579174409699256, 1.64909586407260238552982856705, 2.76027499970656701020258651001, 3.99848833583257034878116758300, 5.20559650209889299858183743130, 6.03167711977964999131298998837, 6.65756814130184733836658355592, 7.977478528957323231564969684504, 8.754290633619139071331480123889, 9.637107458154946260139791366279, 9.969793758600395329514547737780

Graph of the $Z$-function along the critical line